### The Karma of Kaplan-Meier

#### (May 7, 2018)

Our new book, Modelling Mortality with Actuarial Applications, describes several non-parametric estimators of two quantities:

1. The survival function, $$S_x(t)$$, defined as the probability that a person now aged $$x$$ will survive at least $$t$$ years ($${}_tp_x$$ to actuaries), and
2. The integrated hazard function, $$\Lambda_x(t) = \displaystyle\int_0^t\mu_{x+s}ds$$.

The estimators of the above quantities are based on two items of data collected at the times of the observed deaths (denoted by $$t_1,t_2,\ldots,t_n$$):

1. The number, $$d_{x+t_i}$$, who died at time $$t_i$$, and
2. The number, $$l_{x+t_i^-}$$, who were alive and under observation immediately before time $$t_i$$ (which time we denote…

### Spotting hidden data-quality issues

#### (Nov 3, 2013)

The growing market for longevity risk-transfer means that takers of the risk are keenly interested in the mortality characteristics of the portfolio concerned. The first thing requested by the risk-taker is therefore detailed data on the portfolio's recent mortality experience.  This is ideally data extracted on a policy-by-policy basis. Once received, the careful analyst checks that the data are sound.  Failure to spot data problems at the start will at best waste time, and at worst lead to concluding a deal on bad terms.  There is therefore tremendous value in simple checks of data quality.

We saw in an earlier post how survival models can reveal data problems.  However, these issues can sometimes be spotted…

### Early retirements

#### (Mar 25, 2009)

Members of defined-benefit pension schemes can often retire early if they are in poor health.  Unsurprisingly, such ill-health retirements exhibit higher mortality rates than those who retire at the normal scheme age.

Over time, however, the information on the health status of a pensioner is often lost.  When administrators are changed, for example, the original reason for retirement may not be migrated across onto the new payment system.  This poses a dilemma: we know that the reason for retirement will be a material risk factor, but we often won't know the reason codes for all pensioners.

One solution adopted by the CMI in the U.K. is to assume that everyone whose pension began before a certain age has retired…

### Are you allergic to statistical models?

#### (Aug 4, 2008)

Or do you know someone who is? Some people are uncomfortable with the idea of statistical models, especially ones with parameters. It is worth remembering that in 1958 Kaplan and Meier introduced the idea of an empirical survival curve, also called the product-limit estimator. The basic idea is to re-arrange the mortality experience data in such a way as to demonstrate the survival rates of different sub-groups. The key feature of the Kaplan-Meier curve is that there are no parameters involved: the empirical survival curve is simply a re-arrangement of the experience data, and involves no model fitting and no parameter estimation.

In the chart below we show the Kaplan-Meier curves for males and females in a…